Editing Spectrum (functional analysis) (section)

{{Short description|Set of eigenvalues of a matrix}}
{{for|the prime spectrum of a ring|Spectrum of a ring}}
In [[mathematics]], particularly in [[functional analysis]], the '''spectrum''' of a [[bounded operator|bounded linear operator]] (or, more generally, an [[unbounded operator|unbounded linear operator]]) is a generalisation of the set of [[eigenvalue]]s of a [[matrix (mathematics)|matrix]].  Specifically, a [[complex number]] <math>\lambda</math> is said to be in the spectrum of a bounded linear operator <math>T</math> if <math>T-\lambda I</math>

* either has ''no'' set-theoretic [[inverse function|inverse]];
* or the set-theoretic inverse is either unbounded or defined on a non-dense subset.<ref>{{cite book |last1=Kreyszig |first1=Erwin |title=Introductory Functional Analysis with Applications}}</ref>

Here, <math>I</math> is the [[Identity function|identity operator]].

By the [[closed graph theorem]], <math>\lambda</math> is in the spectrum if and only if the bounded operator <math>T - \lambda I: V\to V</math> is non-bijective on <math>V</math>.

The study of spectra and related properties is known as ''[[spectral theory]]'', which has numerous applications, most notably the [[mathematical formulation of quantum mechanics]].

The spectrum of an operator on a [[Dimension (vector space)|finite-dimensional]] [[vector space]] is precisely the set of eigenvalues.  However an operator on an infinite-dimensional space may have additional elements in its spectrum, and may have no eigenvalues.  For example, consider the [[unilateral shift|right shift]] operator ''R'' on the [[Hilbert space]] [[Lp space|ℓ<sup>2</sup>]],
:<math>(x_1, x_2, \dots) \mapsto (0, x_1, x_2, \dots).</math>
This has no eigenvalues, since if ''Rx''=''λx'' then by expanding this expression we see that ''x''<sub>1</sub>=0, ''x''<sub>2</sub>=0, etc.  On the other hand, 0 is in the spectrum because although the operator ''R''&nbsp;−&nbsp;0 (i.e. ''R'' itself) is invertible, the inverse is defined on a set which is not dense in [[Lp space|ℓ<sup>2</sup>]]. In fact ''every'' bounded linear operator on a [[complex number|complex]] [[Banach space]] must have a non-empty spectrum.

The notion of spectrum extends to [[unbounded operator|unbounded]] (i.e. not necessarily bounded) operators. A [[complex number]] ''λ'' is said to be in the spectrum of an unbounded operator <math>T:\,X\to X</math> defined on domain <math>D(T)\subseteq X</math> if there is no bounded inverse <math>(T-\lambda I)^{-1}:\,X\to D(T)</math> defined on the whole of <math>X.</math>  If ''T'' is [[closed operator|closed]] (which includes the case when ''T'' is bounded), boundedness of <math>(T-\lambda I)^{-1}</math> follows automatically from its existence. U

The space of bounded linear operators ''B''(''X'') on a Banach space ''X'' is an example of a [[unital algebra|unital]] [[Banach algebra]].  Since the definition of the spectrum does not mention any properties of ''B''(''X'') except those that any such algebra has, the notion of a spectrum may be generalised to this context by using the same definition verbatim.